Preheat oven to 400 degrees.
Butter a 9-by-9-inch square baking dish or a 10-by-6 1/2-inch oval baking dish with 2 teaspoons butter. Cut a piece of foil large enough to cover the dish and line with parchment paper. Butter parchment paper with 1 teaspoon butter. Set prepared dish, foil, and parchment aside until ready to bake.
In a small bowl, whisk together chicken broth, egg, and egg yolk; set aside.
Melt 1 tablespoon butter over medium-high heat in a medium skillet. Add bacon and cook for 1 minute. Add onion and celery and cook until softened, about 4 minutes more. Increase heat to high and add mushrooms, stirring to combine. Cook, stirring occasionally, for 2 minutes; stir in garlic. Remove skillet from heat and add parsley and sage. Transfer mushroom mixture to a medium bowl and add breadcrumbs, stirring carefully to combine; season with salt and cayenne pepper. Add chicken broth and egg mixture to stuffing, stirring until well combined. Transfer stuffing to prepared baking dish.
Melt remaining 4 tablespoons butter in a small skillet over medium heat until bubbling and lightly browned, about 2 minutes. Drizzle brown butter over stuffing and cover with prepared foil. Set baking dish on a baking sheet and transfer to oven. Bake for 25 minutes, uncover, and continue baking 15 minutes more. Remove from oven and let stuffing cool for 5 minutes before serving.